ECE697AA Lecture 17. Birth-Death Processes

Transcription

1 Tlman Wolf Department of Electrcal and Computer Engneerng /4/8 ECE697AA ecture 7 Queung Systems II ECE697AA /4/8 Uass Amherst Tlman Wolf Brth-Death Processes Solvng general arov chan can be dffcult Smpler, constraned verson: brth-death process Transtons are only allowed between neghborng states Transton rates: brth rate and death rate Brth-death process: atrx form: O Q

2 ECE697AA /4/8 Uass Amherst Tlman Wolf Steady State of Brth-Death Processes Steady state equatons: Solvng for : / / In general:, What about? Sum of probabltes must be Convergence crteron:, > : / < O Q ECE697AA /4/8 Uass Amherst Tlman Wolf 4 Brth-Death Process Example Smplest example All brth rates are the same All death rates are the same Solve : Then : Represent utlzaton ρ/ -ρρ Geometrc dstrbuton wth parameter p-ρ / /

3 Brth-Death Process Example ean number of customers n system: N ρ ρ ρ ρ Wth ttle s law: TN///-ρ Qρ /-ρ So, fnally: Wth ncreasng load, queue length and watng tme ncrease ECE697AA /4/8 Uass Amherst Tlman Wolf 5 Kendall s Notaton There are many dfferent queung systems Notaton ndcates type of arrval and servce Exponental dstrbuton memoryless D Determnstc dstrbuton G General dstrbuton Queung dscplne ndcates Arrval process Servce process Number of servers E.g.: // Smplest case prevous example ECE697AA /4/8 Uass Amherst Tlman Wolf 6

4 // queung model // results: Brth-death process wth and» -ρρ» -ρ Average number of jobs n system» Kρ/-ρ Average response tme» TN/ / -ρ ean queue length» Qρ /-ρ What are the assumptons? Exponentally dstrbuted nterarrval and servce tmes ECE697AA /4/8 Uass Amherst Tlman Wolf 7 /G/ queung model Servce tme s not exponentally dstrbuted What does pacet transmsson tme depend on?» Pacet sze» n speed constant We need dfferent model Generalzed dstrbuton for servce tme How can we model such a servce tme? From pont of vew of arrvng job Watng tme depends on» Remanng servce tme of current job W» Sum of mean servce tmes of jobs n queue Q E[X] Thus, WW Q E[X] ECE697AA /4/8 Uass Amherst Tlman Wolf 8 4

5 /G/ queung model Expected servce tme s ndependently dstrbuted Use ttle s law» WW Q E[X] W W E[X] Wth E[X]/» WW ρ W Solve for W» WW /-ρ What s value of W? Depends f server s busy or not W P[busy] RP[not busy] How can we determne mean resdual lfe R? Result from Klenroc» R/ E[X ]/E[X]/ E[X]c X c X, where c s coeffcent of varaton c X σ X /E[X] normalzed standard devaton ECE697AA /4/8 Uass Amherst Tlman Wolf 9 /G/ queung model Total watng tme: WW /-ρρ/-ρ / E[X]c X Wth ttle s law Q W and E[X]/: ρ cx ρ E[ X ] Q ρ ρ E[ X ] Pollacze-Khntchne formula Santy chec: Exponental dstrbuton for G» σ X /, E[X]/,c X σ X /E[X]» Qρ /-ρ ECE697AA /4/8 Uass Amherst Tlman Wolf 5

6 /D/ queung model Determnstc servce tme Examples Servce of requests» Web page» DNS looup emory access Coeffcent of varaton c X Queung tme Q/ ρ /-ρ ECE697AA /4/8 Uass Amherst Tlman Wolf /G/ // comparson How much do /G/ and // dffer? Assume networ traffc //» Servce tme exponentally dstrbuted /G/» Servce tme proportonal to pacet sze Queue length ρ c ρ /G/ queue shorter f X < ρ ρ Need c X for pacets What s the dstrbuton of pacet lengths? ECE697AA /4/8 Uass Amherst Tlman Wolf 6

7 From NANR: E[X]54 E[X ]5755 σ X 598 c X.687 c X.844 Thus c X // s too optmstc Pacet length dstrbuton.844 > ECE697AA /4/8 Uass Amherst Tlman Wolf Homewor Read Nc ckeown, Pravn Varaya, and Jean Walrand, Schedulng cells n an nput-queued swtch, IEEE Electroncs etters, vol. 9, no. 5, pp , Dec. 99. SPARK Assessment quz ECE697AA /4/8 Uass Amherst Tlman Wolf 4 7